Min-Additive Utility Functions

Transcription

1 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. Mn-Addtve Utlty Fnctons by Brce W. Laar The MITRE Corporaton Bedford, MA Febrary 9 ABSTRACT Ths paper ntrodces the n-addtve also called n-average tlty fncton. Ths fncton s a eghted cobnaton of an addtve tlty fncton and a nzaton over a set of sngle attrbte tlty fnctons. The eghtng s accoplshed by eplotng nforaton already contaned n the addtve and nzaton odels. For fors of the n-addtve MA odel are presented basc, nfor, logstc, and relaed. The basc MA odel generalzes the addtve and nzaton odels bt does not reqre any addtonal paraeters to be estated. It can be eployed n statons here the decsonaker s preferences volate the addtve ndependence assptons nherent n the addtve odel. The nfor MA odel etends the basc MA odel by addng locaton and spread paraeters. The logstc MA odel etends the nfor MA odel by creatng a contnosly dfferentable eghtng fncton. Ths eghtng fncton s shon to be a close approaton of a Gassan clatve dstrbton fncton. The relaed MA odel reoves the non-negatvty reqreents on the eghts. Ths verson of the MA odel s shon to be a generalzaton of the to-densonal lt-lnear tlty fncton and the to-densonal ltplcatve tlty fncton. Nercal eaples and graphcal representatons of the odels are presented. The paper contans three appendces. Append A llstrates ho the MA odel can be nested n a decson preference herarchy. Append B copares the MA odel to the recently proposed lted average and eponental-average faly of tlty fnctons. Fnally, Append C sarzes the copleent to the MA odel the a-addtve odel. The aaddtve odel s sed n rsk analyss and other statons nvolvng dstltes.. Introdcton Let {,,, n} be the set of attrbtes that are of nterest to the decson aker DM. BEST Each attrbte vares fro a least preferred vale to a ost preferred vale. A tlty fncton [Keeney 99, p.3] s a real-valed fncton that epresses the DM s strength of preference for varos levels of. In practce, the ost coonly sed tlty fncton s the addtve odel of the for n ADD

2 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. C MID and MID D BEST and MID Fgre Alternatves A and B Fgre Alternatves C and D here the s are a set of n non-negatve eghts.e., constants that s to nty, the s are a set of n sngle densonal.e., sngle attrbte tlty fnctons and = {,,, n n}. Typcally, each s calbrated so that and BEST. Then ADD also ranges beteen and. There s an etensve and ellestablshed lteratre for deternng the fnctonal for of the s and for assessng the vales of the s n the addtve odel see, for eaple, [Cleen and Relly 4], [Keeney and Raffa 976], [Krkood 997], [Mollaghase and Pet-Edards 997], [Poerol and Barba-Roero ], [Raffa 968], and [von Wnterfeldt and Edards 986]. The addtve odel asses addtve ndependence [Fshbrn 964, pp ], [Keeney and Raffa 976, pp ] eanng, sply, that ADD s assed to be a lnear fncton th respect to the ndvdal tltes,. Ths, ADD /, the argnal rate of change of ADD th respect to, s assed to be constant naely the eght regardless of the vale of any of the ndvdal tltes,,, n. [Cleen and Relly 4, p. 585], [Krkood 997, p. 5], [von Wnterfeldt and Edards 986, p. 39], and others pont ot that addtve ndependence s, n soe cases, an nrealstc asspton. A classc eaple see, for eaple, [Krkood 997, p. 5] llstratng the plcatons of ths asspton nvolves a project anager the DM ho s consderng to attrbtes project cost and project BEST schedle. The cost attrbte can vary fro project copleted n bdget to project sffers nacceptable cost overrns. The schedle attrbte can vary fro BEST project copleted on schedle to project sffers nacceptable delays. We set BEST BEST,,, and. The DM s asked to consder to alternatves A and B see Fgre. For alternatve A there s a 5% chance that the project ll be n bdget bt ll fal de to nacceptable schedle delays, and a 5% chance that the project ll be on schedle bt ll fal de to nacceptable cost overrns. In contrast, for alternatve B, there s 5% chance that the project ll be copleted n bdget and on te, and a 5% chance that the project ll have both nacceptable cost overrns and nacceptable delays.

3 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. As shon n eq., nder the addtve ndependence asspton, the epected tlty for the to alternatves s the sae plyng that the DM s ndfferent beteen the to alternatves: E[ ADD E[ ADD A] B] Hoever, ths ndfference conclson s nrealstc snce t asses that the DM s llng to tradeoff nacceptable otcoes n the sae anner as she/he s llng to tradeoff acceptable otcoes. In realty, f the project s falng becase of nacceptable cost overrns, the DM s ore lkely to have a lo tlty assessent regardless n the stats to the schedle attrbte. Slarly, f the project s eperencng nacceptable schedle delays, the DM ll have a lo tlty assessent rrespectve of the project costs. Ths, hen nacceptable otcoes are beng consdered the DM s strength of preference s ore accrately represented by a nzaton odel of the for MIN n{,,, n } 3 n here, as th eq., the s are the set of n sngle densonal tlty fnctons. Applyng the nzaton odel to the to alternatves depcted n Fgre yelds E[ MIN E[ MIN A] B] n{, } n{,} n{,} n{, } 4 The nzaton odel ndcates that the DM prefers alternatve B th a 5% chance of project sccess to alternatve A th a % chance of project sccess. Ths eaple as specfcally constrcted to llstrate the ltatons of the addtve odel. Bt the nzaton odel has ltatons as ell. To llstrate the shortcongs of the nzaton odel, sppose the DM.e., the project anager s asked to consder another par MID of alternatves C and D see Fgre. Let be the vale of sch that MID ; MID slarly, let be the vale of sch that MID MID. In alternatve C, and MID BEST so that and. On the other hand, n alternatve D, and MID so that and. Note that alternatve D s at least as good as alternatve C n all the attrbtes and strctly better than alternatve C n at least one attrbte. Ths, alternatve D donates alternatve C, plyng that the DM shold prefer alternatve D to alternatve C. The DM s preference for alternatve D s reflected n the addtve odel hch ndcates that the DM old prefer alternatve D as long as ADD : / E[ ADD E[ ADD C] D] 5 3

4 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. In contrast, the nzaton odel ples that the DM old be ndfferent beteen alternatves C and D: E[ MIN E[ MIN C] D] n n { {,, } } 6 Ths dscsson otvates the need for a coposte tlty fncton that captres the desrable aspects of the addtve and nzaton odels hle, at the sae te, avods the ptfalls nherent n each of these to odels. One establshed approach for ncorporatng the propertes of both the addtve and the nzaton odel s the lt-lnear odel [Keeney and Raffa 976, pp ]. Ths odel etends the addtve odel to nclde the cross-prodcts of the ndvdal tlty fnctons. The lt-lnear odel s of the for MULTILIN n j n j n jk j k j n j j n j n j j k k n j j,, n 7 here the s, j s, jk s,,,, n are a set of eghts.e., constants and the s are a set of n sngle densonal tlty fnctons. The frst saton n eq. 7 s the addtve odel. The reanng ters n ths eqaton are cross prodcts of the ndvdal tlty fnctons. If any of the s n a cross-prodct ter s zero, then the vale of that ter s zero. Ths, for nacceptable otcoes, the cross-prodct ters rror the effect of the nzaton odel. Althogh the lt-lnear odel has been sccessflly appled n practce, t has to drabacks. Frst, note that the lt-lnear odel contans n eghts, j, etc.. Ths, the nber of eghts that need to be assessed skyrockets as the nber of ndvdal tlty fnctons ncreases. Second, n any cases the cross-prodct ters do not have a eanngfl nterpretaton to the DM. Other etensons of the addtve odel, sch as the ltplcatve odel and hgher order polynoals, have been proposed [Krantz et. al 97, pp. 3 38]. Hoever, they sffer fro soe of the sae drabacks as the lt-lnear odel. In ths paper, e propose an alternatve approach for eldng the characterstcs of the addtve and nzaton odels. Ths approach cobnes the addtve and nzaton fors n a straghtforard anner, reqres fe addtonal paraeters to be estated, can be vsalzed 4

5 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. graphcally, and has a relatvely easy nterpretaton for the DM and the analyst!. We call ths for of tlty fncton the n-addtve odel. The reander of ths paper s organzed as follos. Secton ntrodces the basc n-addtve odel. Secton 3 etends the basc MA odel by addng locaton and spread paraeters sng a nforly dstrbted eghtng fncton. Secton 4 offers a frther refneent of the basc MA odel sng a logstc eghtng fncton. Ths eghtng fncton closely approates a Gassan clatve dstrbton fncton. Secton 5 proves that a varaton of the nfor MA odel s a generalzaton of the to-densonal lt-lnear odel and the to-densonal ltplcatve tlty fncton. Secton 6 gves a graphcal representaton of the n-addtve odel and provdes a nercal eaple. Secton 7 sarzes the paper. Append A llstrates ho the n-addtve odel can be nested n a decson-akng herarchy. Append B copares the n-addtve odel th the recently proposed lted average [Moynhan and Sh 4] and the eponental average [Schdt 7] odels. Append C descrbes the copleent to the MA odel the a-addtve odel sed to represent the DM s preference strctre n statons nvolvng dstltes.. Basc Mn-Addtve Model Note that both the addtve and the lt-lnear odels se constant coeffcents.e., the eghts. Hoever, f the eghts to be assessed n a tlty fncton are pertted to be fnctons of the attrbtes rather than constants, then the dstncton beteen a eght and a tlty blrs. They both are fnctons of the attrbtes, they both vary beteen and, and they both epress the preferences of the DM. Ths, perttng the eghts to be fnctons actally eases the brden of developng a tlty fncton snce the tltes theselves can serve as eghts. In partclar, consder a eghted cobnaton of the addtve and nzaton odels of the for MA MIN ADD 8 MIN here ADD and MIN are defned n eqs. and 3, respectvely, and ADD and MIN are non-negatve eght fnctons that s to one. The net qeston s: What shold be the fnctonal for of the eghts ADD and MIN? Bt, n prncpal, e have already ansered ths qeston. For nstance, as e observed n the Introdcton, the portance of the MIN odel ncreases as the n vale of the ndvdal tlty fnctons.e., the s approaches zero. Bt, the n vale of the ndvdal tlty fnctons s sply MIN tself. Ths, the nforaton abot the vale of the eght of MIN can be nferred drectly fro MIN, or ore precsely, fro the copleent of In prevos ork, e sed the label n-average odel. [Laar and Schdt 4]. Ths, n-addtve and n-average are synonys for the sae odel. ADD 5

6 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. MIN. In other ords, MIN MIN. Frtherore, snce the eghts n eq. 8 s to one, e have MIN. Ths, the basc for of the n-addtve odel s ADD MA MIN MIN MIN ADD 9 here ADD s defned n eq. and MIN s defned n eq.3. The sbscrpt attached to MA s sed to denote the basc for of the n-addtve odel. Three etensons of the odel denoted MA, MA and MA 3 are presented n Sectons 3 throgh 5. To llstrate the perforance of the basc n-addtve odel, e apply ths odel to the for alternatves A, B, C, and D depcted n Fgres and. E[ MA E[ MA E[ MA E[ MA A] B] C] D] 4 The calclatons n eq. sho that, accordng to the basc n-addtve odel, alternatve B s preferred to alternatve A. Ths preference s reasonable snce alternatve B has a 5% chance of project sccess, hereas alternatve A has a % chance of project sccess. The n-addtve odel also ndcates that the DM s ndfferent beteen alternatves B and C. In addton, the n-addtve odel ndcates that alternatve D s preferred to alternatve C. Ths preference s reasonable snce alternatve D donates alternatve C. In coparng eq. th eq. 5, e see that the tlty vale of alternatve D s loer n the n-addtve odel than t s n the addtve odel 4 copared to. The naddtve tlty vale s loer becase ths odel dsconts the tlty vale of an alternatve based on the proty of the attrbtes assocated th that alternatve. As the attrbte vales get closer and closer to an nacceptable vale, the dscontng of the tlty vale gets greater and greater. Ths, fro a atheatcal prograng perspectve, the n-addtve odel can be veed as a conve cobnaton of a tlty azng objectve fncton and a barrer fncton [Bazaraa and Shetty 979, pp ] constranng the solton space to a feasble regon. Ths vepont also enables s to refne the defnton of sch that. Let be the vale of attrbte beyond hch the solton s nfeasble to the DM. In the project anageent eaple gven above, as defned as an nacceptable cost overrn and as defned as an nacceptable schedle delay. That eaple pled dre 6

7 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. conseqences f the vale of as eceeded. Hoever, that need not be the case. For eaple, [von Wnterfeldt and Edards 986, p. 36] descrbe the faclty locaton proble of an eectve the DM n a consltng fr ho ants to open a branch offce n sothern Calforna. One attrbte of DM s locaton decson s the dstance of the offce fro LAX nternatonal arport. In ths eaple, the DM s nllng to consder stes that are ore than one-hor s drvng te fro LAX. Hence, 6 ntes for the drvng te attrbte,. There s nothng rong, per say, th offces located ore than an hor aay fro LAX. They are jst not feasble soltons to the DM s decson proble at hand. The net three sectons present three etensons of the basc n-addtve odel. MA ADD MIN MA MA Fgre 3 Probablty nterpretaton of MA eght fncton 3. Unfor Mn-Addtve Model Let MIN and let MA be the eght appled to the ADD ter n eq. 8. We call MA the n-addtve eght fncton or MA eght fncton, for short. By sbstttng the MA eght nto eq. 8, ths eqaton can be re-rtten as MA MIN ADD MA For any gven, MA can be nterpreted as the condtonal probablty that the DM prefers the addtve odel rather than the nzaton odel; and the vale of the MA tlty fncton can be nterpreted as the condtonal epected vale of these to odels see Fgre 3. Let MA denote the vale of the MA eght, MA, n the basc n-addtve odel gven n eq. 9. The vale of MA ncreases nforly fro to as vares fro to. That s, f f MA MA f In fact, MA s the clatve dstrbton fncton CDF of a nforly dstrbted rando varable n the doan [,] see Fgre 4. Approately les drng rsh-hor. 7

8 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. MA / MA a Weght fncton b Dervatve of eght fncton Fgre 4 Basc MA eght fncton MA / MA r r a Weght fncton b Dervatve of eght fncton Fgre 5 Unfor MA eght fncton MA / MA % % r r a Weght fncton b Dervatve of eght fncton Fgre 6 Logstc MA eght fncton 8

9 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. In order to etend the basc n-addtve odel, let be the vale of belo hch the DM eclsvely accepts the nzaton odel; and let r be the vale of above hch the DM eclsvely accepts the addtve odel. In the basc n-addtve odel, the left endpont,, s fed at and the rght endpont, r, s fed at. A natral etenson of the basc n-addtve odel s to allo and r to be paraeters. We call ths etenson the nfor n-addtve odel hose tlty fncton vale s denoted by MA and hose MA eght fncton s denoted by MA. In the nfor n-addtve odel, MA s the CDF of a nforly dstrbted rando varable n the doan, ] see Fgre 5. Specfcally, [ r MA MIN ADD 3 MA MA here ADD and MIN are defned n eqs. and 3, respectvely, and MA s gven by f MA f 4 f here s the d-pont beteen and r, and s the dfference beteen and r see Fgre 5. That s, Invertng eq. 5 gves r and r 5 and r 6 Note that s a easre of locaton and s a easre of spread of the dstrbton. Ths, varyng shfts the dstrbton to the left or rght along the real nber lne; and varyng alters the dstance beteen the left and rght endponts, and r. Eqvalently,.e., the recprocal of s eqal to the a rate of change of MA th respect to, and ths s a easre of steepness of the dstrbton. These paraeters can be set to represent the preferences of the DM. For eaple, f. 5 and.4, sng eq. 6 gves. 3 and r. 7. These paraeter vales ndcate that the DM strctly prefers the nzaton odel eq. 3 f any of the sngle densonal tlty fncton.e., the s has a vale belo.3. Conversely, the DM strctly prefers the addtve odel eq. f all of the vales are above.7. Otherse, the DM prefers a eghted cobnaton of the addtve odel and the nzaton odel th the eght gven by eq. 4. 9

10 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. Note that the paraeters and r are not reqred to be beteen and. The only restrcton placed on these to paraeters s that r. Ths gves a far aont of fleblty n calbratng the nfor n-addtve odel to reflect the DM s preferences. In partclar, If r and r, then MA ADD 7a If and r, then MA MIN 7b The net secton presents a frther etenson of the basc n-addtve odel. 4. Logstc Mn-Addtve Model The nfor n-addtve odel dscssed n the prevos secton ses a pecese-lnear MA eght fncton see Fgre 5 and eq. 4. As noted, the paraeters and r or correspondngly and can be adjsted to reflect a range of DM preferences. Bt, despte these advantages, the pecese-lnear fnctonal for also has to drabacks one practcal and the other atheatcal. Frst, as a practcal atter, the DM ay be ncofortable or nable to specfy a pont belo hch he/she eclsvely prefers the nzaton odel, or a pont r above hch he/she eclsvely prefers the addtve odel. Second, the pecese-lnear fncton gven by eq. 4 s not dfferentable at the endponts and r. These to drabacks arge for replacng MA th a sooth fncton that asyptotcally approaches zero resp. one as the vale of decreases resp. ncreases hle, at the sae te, retans the desrable characterstcs of MA. A convenent fncton satsfyng these reqreents s a logstc fncton.e., a sgod fncton or S-shaped fncton. Hence, e refer to ths etenson as the logstc n-addtve odel. The logstc n-addtve tlty fncton, denoted as MA, s specfed as MA MIN ADD 8 MA MA here ADD and MIN are defned n eqs. and 3, respectvely, and MA s gven by a logstc fncton 3 of the for see Fgre 6 MA 9 ep 4 3 The ore coon for of logstcs fncton s ep The coeffcent 4 s sed n eq. 9 n order to keep the notaton sed n the logstc n-addtve odel consstent th the notaton sed n the other versons of n-addtve odel.

11 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. here ep denotes eponentaton of the Eler nber, e. The paraeters and n the logstcs MA eght fncton play analogos roles to the paraeters and n the nfor MA eght fncton see eq. 4. Naely, s a easre of locaton and s a easre of spread. For the logstc MA eght fncton, e defne to addtonal paraeters, denoted and r, here s the pont belo hch DM strongly prefers 4 the nzaton odel MIN ; and r s the pont at hch DM strongly prefers 5 the addtve odel ADD. Then s defned as the d-pont beteen and r ; and s defned as the dfference beteen and r. Ths, the follong relatonshps hold: r, r,, r These relatonshps enable s to descrbe the phrase strongly prefers sed n the defnton of and r ore precsely. Specfcally, sbstttng, respectvely, and r for n eq. 9 yelds MA.88 and r.88 MA e e Ths, at, there s an 88% chance that the DM ll prefer the nzaton odel hereas, at r, there s an 88% chance that the DM ll prefer the addtve odel. Moreover, agan analogos th the nfor n-addtve odel,.e., the recprocal of s a easre of steepness of the dstrbton becase s eqal to the a rate of change of MA th respect to. To sho ths steepness property, note that the dervatve of MA th respect to s gven by MA / cosh 4 here cosh denotes the hyperbolc cosne fncton. Observe that cosh and that cosh y for all y. Applyng the hyperbolc cosne propertes to eq. shos that the a vale of MA / s attaned at and that MA /. Frtherore, these propertes sho that MA / ests for all n the doan, provded that. Ths condton on, n trn, ples that the only reqreent on and r s that r. Ths, as th the nfor n-addtve odel, there s a great deal of fleblty n settng the paraeters of the odel. 4 Defned belo. 5 Ibd.

12 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. MA r r a Logstc MA eght fncton b Gassan CDF Fgre 7 Coparson of the logstc MA eght fncton and the Gassan CDF Fnally, Craer 3 and others have noted the slarty beteen a logstc fncton and a Gassan.e., Noral clatve dstrbton fncton CDF see Fgre 7. Let denote a Gassan CDF and let / be a Gassan densty fncton gven by ep 3 here s the ean, s the standard devaton, and ep denotes eponentaton of the Eler nber, e. Note that s also the pont of syetry and that, at, e have / /.5. Eqatng eqs. and 3 at ther pont of syetry yelds r r and In other ords, the logstc MA eght fncton can be approated by the CDF of a Gassan dstrbton th a ean of and a standard devaton of approately 5. The paraeters of the n-addtve odel can also be set to represent a to-densonal ltlnear odel. Ths verson of the n-addtve odel s dscssed net. 5. Relaed Mn-Addtve Model As noted n Secton 3, the addtve and nzaton odels are specal cases of the nfor n-addtve odel see eq. 7. In ths secton e sho that a to densonal.e., to attrbte lt-lnear odel s also a specal case of the n-addtve odel. The general n-densonal for of the lt-lnear odel as gven n eq. 7. The to-densonal verson [Keeney and Raffa 976, pp ] of ths eqaton s

13 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. MULTILIN 5 here,, and are eghts.e., constants. and and are sngle densonal.e., sngle attrbte tlty fnctons. In the lt-lnear odel, the frst to eghts, and, are non-negatve constants; and the three eghts s to one. That s, =. If eqals zero, then the lt-lnear odel reverts to a pre addtve odel and, as shon n Secton 3, the nfor n-addtve odel s eqvalent to the addtve odel f r and r see eq. 7. On the other hand, f, then ths eght s pertted to take on negatve as ell as postve vales. Ths relaaton s eqvalent to perttng the nfor MA-eght fncton, MA, see eq. 4 to take on vales otsde the range [,]. We let MA 3 denote the relaed verson of the MA eght fncton here MA s sply see Fgre 8 3 MA 6 3 Ths verson of the odel, called the relaed n-addtve odel, s denoted by MA 3 and s specfed by MA MIN ADD 7 3 MA 3 MA 3 here ADD and MIN are defned n eqs. and 3, respectvely, and MA 3 s gven by eq. 6. Coparng eq. 4 n Secton 3 th eq. 6 n ths secton, e see that the nfor MA eght fncton, MA, s coposed of three pecese-lnear segents hereas the relaed MA eght fncton, MA 3, s coprsed of a sngle lnear segent agan, see Fgre 8. MA MA3 r r a Unfor MA eght fncton b Relaed MA eght fncton Fgre 8 Coparson of nfor and relaed MA eght fnctons 3

14 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. 4 Sbstttng the rght-hand-sde of eq. 6 for 3 MA n eq. 7 and epandng the eqaton yelds 3 MIN ADD MIN MIN ADD MIN MA 8 To cast eq. 8 n the for of the lt-lnear odel gven n eq. 5, e consder for possble cases dependng on hether and/or eqals zero. Note that the case here eqals zero has already been covered. Ths, n Case throgh Case 4 belo, e asse that. Case : and, If all three eghts are non-zero, then e ake the follong assgnents: and ADD MIN 9a and 9b Sbstttng the epressons n eq. 9 nto the n-addtve odel specfed n eq. 8 gves 3 MULTILIN MA 3 Case : and, If eqals zero bt the other eghts are non-zero, then e ake the follong assgnents: and ADD MIN 3a and 3b Sbstttng the epressons n eq. 3 nto the n-addtve odel specfed n eq. 8 gves

15 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. 5 3 MULTILIN MA 3 Case 3: and, If eqals zero bt the other eghts are non-zero, then e ake the follong assgnents: and ADD MIN 33a and 33b Sbstttng the epressons n eq. 33 nto the n-addtve odel specfed n eq. 8 gves 3 MULTILIN MA 34 Case 4: and, Fnally, f both and eqal zero, then e ake the follong assgnents: and ADD MIN 35a and 35b Sbstttng the epressons n eq. 35 nto the n-addtve odel specfed n eq. 8 gves 3 MULTILIN MA 36 By consderng these for cases, e have shon that any to densonal lt-lnear odel can be converted nto the relaed n-addtve odel. Hence, ths verson of the n-addtve

16 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. odel nherts all the propertes assocated th the lt-lnear odel e.g., connectvty, transtvty, tal tlty ndependence, etc. [Keeney and Raffa 976, pp. 3 4]. Moreover, note that n Case 4 above, the relaed n-addtve odel as eqated to a todensonal ltplcatve odel. Ths, ths verson of the n-addtve odel s a generalzaton of both the lt-lnear tlty fncton and the ltplcatve tlty fncton. 6. Graphcal Representaton and Nercal Eaple Ths secton provdes a vsal representaton of the odels presented n ths paper. The ateral s dvded nto three parts. Secton 6. graphcally copares the basc MA odel th the addtve and nzaton odels. Secton 6. llstrates the effects of the locaton and spread paraeters sng the nfor and logstc MA odels. Secton 6.3 vsally deonstrates the eqvalence beteen the to densonal lt-lnear odel and the relaed MA odel. [Addtonal graphcal representatons are contaned n Append B.] 6. Addtve, Mnzaton, and Basc Mn-Addtve Models To vsalze the basc for of the n-addtve odel, consder agan the eaple of a project anager the DM ho s consderng to attrbtes project cost and project schedle BEST. The cost attrbte can vary fro project copleted n bdget to project BEST sffers nacceptable cost overrns; and the schedle attrbte can vary fro project copleted on schedle to project sffers nacceptable delays. We set, BEST BEST,, and. For ths eaple, e asse that the to sngle densonal tlty fnctons are eghted eqally.e., and. Fgres 9,, and sho topographc projectons of, respectvely, the addtve odel eq., the nzaton odel eq. 3, and the basc n-addtve odel eq. 9. Each fgre contans a an overhead ve.e., plan ve shong contors projected onto the plane; and b a perspectve ve shong an soetrc sketch contaned thn the nt-cbe. The color bands n the fgres represent ntervals of tlty vales. The contor lnes on the bondary beteen adjacent color bands represent so-tlty crves.e., ndfference crves. The legend for the color bands s sarzed n Table. Color Satch Utlty vale nterval Ble [.8,.] Green [.6,.8] Yello [.4,.6] Orange [.,.4] Red [.,.] Table Legend for topographc contors 6

17 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. a Overhead ve b Perspectve ve Fgre 9 Graphc representaton of addtve odel a Overhead ve b Perspectve ve Fgre Graphc representaton of nzaton odel a Overhead ve b Perspectve ve Fgre Graphc representaton of basc n-addtve odel 7

18 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. Coparng Fgre th Fgres 9 and, e see that the basc MA resebles the addtve odel f both ndvdal tlty vales are n the proty of one. On the other hand, the shape of the basc MA odel s slar to the nzaton odel f ether one or both of the ndvdal tlty vales s n the proty of zero. The topographc projectons can also be sed to vsalze the for alternatves dscssed n Sectons and see Fgres and. Fgre plots the coordnates of alternatves A, B, C, and D on the plane. A B C B D A Fgre Coordnates of alternatves A, B, C, and D ADD ADD ADD D C 3 4 a Alternatve A b Alternatve B c Alternatves C and D Fgre 3 Vale of alternatves A, B, C, an D sng addtve odel MIN MIN MIN C D a Alternatve A b Alternatve B c Alternatves A and D Fgre 4 Vale of alternatves A, B, C, an D sng nzaton odel 8

19 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. MA MA MA C D 5 8 a Alternatve A b Alternatve B c Alternatves C and D Fgre 5 Vale of alternatves A, B, C, an D sng basc n-addtve odel Fgre 3 dsplays the topographc projecton of these for alternatves based on the addtve odel. Ths fgre shos that the addtve odel gves the sae tlty vale for alternatve A and B. Fgre 4 dsplays the topographc projecton of the for alternatves based on the nzaton odel. Ths fgre shos that the nzaton odel does not dstngsh beteen alternatves C and D. Fgre 5 dsplays the topographc projecton of the for alternatves sng the basc MA odel. Ths fgre shos that the basc MA odel represents the DM s preference for alternatve B over alternatve A as ell as the DM s preference for alternatve D over alternatve C. 6. Logstc Md-Addtve and Unfor Mn-Addtve Models Ths secton llstrates the effects of the locaton and steepness paraeters sng the sae project anageent eaple that as sarzed n Secton 6.. The topographc ves shon n ths secton are based on the logstc MA odel. Very slar contors are prodced by the nfor MA odel. Fgres 6 throgh 8 llstrate the effects of alternatng the locaton paraeter,, ceters parbs. Specfcally, the steepness paraeter, s held fed at hle s vared fro to +. At see Fgre 6, the MA odel s alost dentcal to the addtve odel see Fgre 9, at see Fgre 8, the MA odel s very slar to the nzaton odel see Fgre ; and at see Fgre 7, the MA odel s a tre of the to etrees represented by the addtve and nzaton odels. Fgres 9 throgh sho the effects of alternatng the steepness paraeter,, ceters parbs. Specfcally, the locaton paraeter, s held fed at hle s vared fro. to. At. see Fgre 9, the MA eght fncton has a very gentle gradent. A gentle gradent eans that the eght placed on the addtve odel s jst slghtly less than for lo vales of ; and jst slghtly ore than for hgh vales of. In contrast, at see Fgre, the MA eght fncton has a very steep gradent near. A steep gradent eans that the eght placed on the addtve odel changes abrptly for vales of near th alost no eght placed on the addtve odel for and alost % eght placed on the addtve odel for. At., see Fgre the logstc and nfor MA odel perfors n a anner slar to the basc MA odel copare Fgres and. 9

20 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. MA a MA eght fncton b Overhead ve c Perspectve ve Fgre 6 Logstc MA odel th and.e., MA a MA eght fncton b Overhead ve c Perspectve ve Fgre 7 Logstc MA odel th and.e., MA a MA eght fncton b Overhead ve c Perspectve ve Fgre 8 Logstc MA odel th and.e.,

21 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. MA a MA eght fncton b Overhead ve c Perspectve ve Fgre 9 Logstc MA odel th and..e., 5 MA a MA eght fncton b Overhead ve c Perspectve ve Fgre Logstc MA odel th and.e., MA a MA eght fncton b Overhead ve c Perspectve ve Fgre Logstc MA odel th and.e.,

22 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. a Shortage attrbte b Otdatng attrbte Fgre Sngle attrbte tlty fnctons for blood bank eaple 6.3 Mlt-Lnear Model and Relaed Mn-Addtve Model Ths secton llstrates the eqvalence beteen the to densonal lt-lnear odel and the relaed n-addtve odel sng an eaple nvolvng polcy decsons for nventoryng hole blood, blood plasa, and other coponents at a hosptal blood bank [Jennngs 968 pp ], [Keeney and Raffa 976, pp. 73 8], [Cleen and Relly 4, pp ]. Snce deand for blood nts s stochastc, a safety stock st be antaned to nze the probablty of shortages. Yet, blood nts are also pershable and st be dscarded f ther alloable shelf-lfe s eceeded. 6 Let.be the annal percent of nts deanded bt not n stock; and let be the annal percent of nts reoved fro nventory de to otdatng. The nrse n charge of antanng blood spples at the hosptal the DM has deterned that BEST.e., a % shortage rate and.e., no shortage of blood nts. Also, BEST.e., a % dscard rate and.e., no nts dscarded de to otdatng. Her sngle attrbte tlty fnctons ere assessed as e e e BEST.3 e e e e BEST.4 e e e a 37b here e s the Eler nber.e., e The coeffcents.3 and.4 ndcate a greater dstlty for shortages than for otdatng see Fgre. Assng tal tlty ndependence [Keeney and Raffa 976, pp ], a to-densonal lt-lnear odel see eq. 5 as calbrated th. 74,. 38, and Usng these eghts and sbstttng and fro eq. 37 nto eq. 5 yelds 6 Ma alloable storage te for ost blood coponents s approately three eeks.

23 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. MULTILIN e e e e A topographc projecton of eq. 38 s prodced n Fgre 3. Fgre 3a shos the contor lnes for the tlty fncton projected onto the attrbte plane; and Fgre 3b s a 3-densonal renderng of ths fncton. Observe that the tlty fncton decreases as ether or decreases; and that the rate of decrease s greater for than for reflectng the DM s greater concern for shortages of blood rather than for the otdatng of blood. a Overhead ve b Perspectve ve Fgre 3 Mlt-lnear odel for blood bank eaple a Overhead ve b Perspectve ve Fgre 4 Relaed n-addtve odel for blood bank eaple 3

24 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. To constrct the eqvalent representaton sng the relaed n-addtve odel, note that all the eghts n eq. 38 are non-zero. Hence, Case n Secton 5 apples and e set MIN ADD e e e By sbstttng the epressons n eq. 39 nto the relaed n-addtve odel specfed n eq. 8 e get MA e e e e e e e e Coparng eq. 38 th eq. 4, e see that the to eqatons are dentcal. Hence, MA 3 MULTILIN. Ths eqvalence s also confred by coparng the topographc projecton of eq. 38 see Fgre 3 th the topographc projecton of eq. 4 see Fgre 4. The net secton sarzes the paper. 7. Sary Ths paper has presented a faly of n-addtve MA tlty fnctons that generalze the addtve and nzaton tlty fnctons. The MA odels can be eployed n statons here the decson-aker s preferences volate the addtve ndependence assptons nherent n the addtve tlty odel. The basc verson of the MA odel does not reqre any addtonal paraeters to be estated. Etensons of the basc odel se locaton and spread paraeters to specfy a de range of decson-akers preferences. Moreover, these paraeters 4

25 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. can also be set to represent the to-densonal lt-lnear tlty fncton and the todensonal ltplcatve tlty fncton. Ths, the MA tlty odel s a generalzaton of a host of other tlty odels. Etensons of the MA odel presented n ths paper are also possble. Append A shos ho the MA tlty fncton can be nested n a decson tree herarchy; Append B copares the MA odel th to other odels the lted average tlty fncton and the eponentalaverage tlty fncton; and Append C descrbes the a-addtve fncton sed to odel rsk nzaton rather than tlty azaton. For frther eaples of analytcal ethods for rsk anageent, see, for eaple, [Garvey ], [Garvey 9]. In s, these etensons, together th the for versons of the n-addtve odel presented n ths paper can help decson-akers ake better decsons. Append A Ths append llstrates ho the n-addtve odel can be nested n a decson-akng herarchy. Consder the follong hypothetcal staton. 7 Mr. Taylor s classroo has three stdents n t: Xavar X, Yvonne Y, and Zachary Z. Each stdent s takng for sbjects: ENGLISH, HISTORY, MATH, and SCIENCE. Sppose that the stdents have receved the grades for the for sbjects that are posted n Table. Stdent X Y Z Sbject and grade ENGLISH HISTORY MATH SCIENCE Method of Sarzng Grade pont average Loest grade Basc Mn- Addtve odel Table Hypothetcal sbject grades and sary easres for three stdents Ho shold a stdent s grades be aggregated? The tradtonal ethod, of corse, s to copte a grade pont average assgnng 4 grade ponts for an A, 3 grade ponts for a B, etc., sng the grade ponts, and dvdng by the nber of sbjects. The grade pont average sary s shon n Table. The grade pont average sary dstngshes beteen stdents X and Y bt does not dstngsh beteen stdents Y and Z. In partclar, the grade pont average sary does not reflect stdent Z s falng grade n SCIENCE. One ethod of captrng ths falng grade n a sary easre s to report the loest.e., n grade for each stdent. Table also 7 Ths eaple s ntended to llstrate the ork-breakdon-strctre WBS typcal of any syste-of-systes confgratons. It s not ntended to advocate any polces regardng edcatonal assessent. 5

26 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. shos the loest grade sary easre. Ths easre dstngshes beteen stdent Y and stdent Z bt t does not dstngsh beteen stdent X and stdent Y. The basc n-addtve MA sary easre descrbed n Secton dstngshes aong all three stdents. Specfcally, the grade pont average sary easre s copted sng ADD see eq., the loest grade sary easre s specfed sng MIN see eq. 3; and the basc n-addtve MA sary easre s calclated sng MA see eq. 9. As shon n Table, for stdents that are perforng ell gettng A s and B s, the MA sary easre places ephass on the grade pont average. Hoever, f a stdent s recevng a falng grade n any sbject, that nforaton s not overlooked. Ths sae ethod of averagng hgh scores hle red flaggng lo scores can be pleented at ltple levels as llstrated by the hypothetcal herarchy depcted n Fgre 5. The Regonal School node R has to classroos: Mr. Taylor s classroo node T and Ms. Sth s classroo node S. As entoned before, Mr. Taylor s classroo has three stdents n t: Xavar X, Yvonne Y, and Zachary Z see Table. Ms. Sth s classroo also has three stdents n t: Ulysses U, Vctor V, and Wlbert W; and each stdent n Ms. Sth s class s also takng for sbjects: ENGLISH, HISTORY, MATH, and SCIENCE. Startng at the botto level of the tree, the basc MA sary easre s appled to each sccessvely hgher level. Ths, the stdents scores are based on ther corse grades, teachers scores are based on the scores of the stdents n ther class, the school s score s based on the scores of the teachers n the school, and so on. By sng the MA sary easre, overall perforance s easred at each level thot attenatng a falng score at a loer level. The easre also provdes a trace to the root case of a falng score a rghtost depth frst search for the tree herarchy shon n Fgre 5. Moreover, the MA perforance easre can pnpont here addtonal resorces or other correctve easres can ganflly be eployed. R S T U V W X Y Z Fgre 5 Hypothetcal eaple of herarchy of n-addtve perforance easres 6

27 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. Append B Ths append sarzes other ethods that cobne the addtve and nzaton odels. To prosng odels of ths type are the lted average odel [Moynhan and Sh 4] and the eponental average odel [Schdt 7]. B. Lted Average Model The lted average odel, denoted LIMAVG, s gven by LIMAVG n{ ADD, MIN } 4 here ADD and MIN are specfed n eqs. and 3, respectvely, and s a paraeter. To llstrate the affect of, the lted average odel as appled to the sae project anageent eaple descrbed n Secton 6. The topographc projectons of eq. 4 are shon n Fgres 6 throgh 8. The legend for these fgres s sarzed n Table n Secton 6. Fgre 6 shos that hen the lted average odel takes the for of the addtve odel. Fgre 7 shos that hen., the lted average odel s a of the addtve odel and the nzaton odel. Fgre 8 shos that hen, the lted average odel takes the for of the nzaton odel. B. Eponental Average Model The eponental average odel, denoted EXPAVG, s gven by n a a EXPAVG log 4 here a s a paraeter, log a {} the base a logarth, the s are a set of n non-negatve eghts.e., constants that s to nty, and the s are a set of n sngle densonal.e., sngle attrbte tlty fnctons To llstrate the affect of the paraeter a, the eponental average odel as appled to the sae project anageent eaple descrbed n Secton 6. The topographc projectons of eq. 4 are shon n Fgres 9 throgh 3. The legend for these fgres s sarzed n Table n Secton 6. Fgre 9 shos that hen a. the eponental average odel s a tre of the addtve odel and the nzaton odel. Fgre 3 shos that as a asyptotcally approaches one, the eponental average odel takes the for of the addtve odel. Fgre 3 shos that hen a, the eponental average odel s a of the addtve odel and a azaton odel gven by a{,,, n n}. In fact, as a approaches zero, EXPAVG approaches a pre nzaton odel; and as a the recprocal of a approaches zero, EXPAVG approaches a pre azaton odel. For addtonal coparsons of the eponental average, the lted average, and the n-addtve odels, see Schdt 7. 7

28 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. a Overhead ve b Perspectve ve Fgre 6 Lted average odel th a Overhead ve b Perspectve ve Fgre 7 Lted average odel th. a Overhead ve b Perspectve ve Fgre 8 Lted average odel th 8

29 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. a Overhead ve b Perspectve ve Fgre 9 Eponental average odel th a. a Overhead ve b Perspectve ve Fgre 3 Eponental average odel th a a Overhead ve b Perspectve ve Fgre 3 Eponental average odel th a 9

30 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. Append C Ths append sarzes the copleent of the n-addtve odel, naely the a-addtve odel. 8 The a-addtve odel s ntended for decson envronents n hch the decson aker DM shes to nze a dstlty fncton rather than aze a tlty fncton. Ths staton arses freqently n rsk analyss here the DM st prepare for a collecton of possble otcoes that have ndesrable conseqences. For eaple, drng the hrrcane season, eergency responders st plan for potental hgh nds, ncleent eather, rsng ater levels, and a host or other ncertan rsk events, hch, n trn, can trgger other ndesrable otcoes. These rsk events are dstltes. Typcal attrbtes assocated th a rsk event are the probablty of occrrence; and f the event does occr, the te ntl the occrrence, the draton of the occrrence, the pact or severty, etc. see, for eaple, [Garvey ], [Garvey 9]. A dstlty fncton s the copleent of a tlty fncton. That s, lo vales of the dstlty fncton are ore desrable to the DM than hgh vales. Let d be the -th sngle densonal.e., sngle attrbte dstlty fncton for attrbte. For eaple, ght be the nber of eeks ntl a rsk event occrs f t does occr, and d ght be of the for d ep here ep denotes eponentaton of the Eler nber, e, and s a te-constant paraeter. For lo vales of ths fncton hen the event, f t occrs, s any eeks aay, the DM ay be llng to ake tradeoffs to tgate other rsk events. On the other hand, f the vale of d s close to one ndcatng a possbly nent event, the DM ay be less llng to consder tradeoffs. Ths asyetry n the DM s atttde toards tradeoffs s not reflected n an addtve dstlty odel. Hoever, t s captred n the a-addtve odel. There are for versons of the a-addtve odel, copletely analogos to the for versons of the n-addtve odel. Let MAXADD k d denote the k-th verson of the a-addtve odel here the nde k for the basc verson, k for the nfor verson, k for the logstc verson, and k 3 for the relaed verson; and d = { d, d,, d n n } s a set of n sngle densonal.e., sngle attrbte dstlty fnctons. Each d ranges fro zero to one th zero ndcatng the ost preferable vale to the DM and ndcatng the least preferable. The for of the k-th verson of the a-addtve odel s gven by MAXADD k d MA M MAX d MA M ADD d 43 k k 8 The a-addtve odel s also called the a-average odel. The pref a n the a-addtve odel s nderlned to help to vsally dstngsh t fro the pref n n the n-addtve odel. 3

31 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. here ADD d s the addtve odel see eq. th the argent replaced by d, MA k M s the k-th n-addtve eghtng fncton see eqs., 4, 9, and 6, th the argent replaced th M, and M and MAX d are defned as M MAX d a{ d, d,, d n } 44 n Ths strctre creates a very straghtforard relatonshp beteen the correspondng versons of the a-addtve and the n-addtve odels. To specfy ths relatonshp, let Then d for, n 45, MAXADD d MA for k,,4 46 k k here MAXADD k d s defned n eq. 43, MA k s defned n eqs. 9, 3, 8, and 7, d = { d, d,, d n n } s a set of n sngle densonal.e., sngle attrbte dstlty fnctons, and = {,,, n n}. s the coplentary set of n sngle densonal tlty fnctons see eq. 45. To prove eq. 46, e note that, by constrcton, the follong relatonshps hold: MAX d MIN 47a ADD d ADD 47b here MAX d s defned n eq. 44, MIN s defned n eq. 3 and ADD d and ADD are defned n eq.. Note also that snce MIN and M MAX d, eq. 47a can be rertten as Sbstttng eqs.47 and 48 n eq. 43 yelds the follong dentty: M 48 MAXADD k d MA k k MIN MA k MA k ADD MIN MA MA k MA k MA k M MAX d MA k MA k MA k M ADD d ADD MIN MA k MA k ADD 49 Eq. 49 shos that the a-addtve odel s the copleent of the n-addtve odel and ths shares the propertes nherent n the n-addtve faly of odels. 3

32 Ths docent has been approved for pblc release. Case nber Dstrbton nlted. B.W. Laar, Mn-Addtve Utlty Fnctons, pp The MITRE Corporaton. All rghts reserved. References [] M.S. Bazaraa and C.M. Shetty 979, Nonlnear Prograng: Theory and Algorths,Wley and Sons, Ne York, NY. [] R.T Cleen and T. Relly 4, Makng Hard Decsons Wth Decson Tools, Dbry Press, Belont, CA. [3] J.S. Craer 3, Logt Models fro Econocs and Other Felds, Cabrdge Unversty Press, Cabrdge, England. [4] P.C. Fshbrn 964, Decson and Vale Theory, Wley and Sons, Ne York, NY [5] P.R. Garvey, Rsk Manageent, Encyclopeda of Electrcal and Electroncs Engneerng, Wley and Sons, Ne York, NY. [6] P.R. Garvey 9, Analytcal Methods for Rsk Manageent: A Systes Engneerng Perspectve, Chapan and Hall/CRC-Press, Ne York, NY. [7] J.B. Jennngs 968, An analyss of hosptal blood bank hole blood nventory control polces, Transfson, vol. 8, no. 6, pp [8] R.L. Keeney 99, Vale-Focsed Thnkng: A Path to Creatve Decsonakng, Harvard Unversty Press, Cabrdge, MA. [9] R.L. Keeney and H. Raffa 976, Decsons th Mltple Objectves: Preferences and Vale Tradeoffs, Wley and sons, Ne York, NY. [] C.W. Krkood 997, Strategc Decson Makng: Mltobjectve Decson Analyss th Spreadsheets, Dbry Press, Belont, CA [] D.H. Krantz, R.D. Lce, P. Sppes, and A. Tversky 97, Fondatons of Measreent: Vole I Addtve and Polynoal Representatons, Acadec Press, Ne York, NY [] B.W. Laar and B.K. Schdt 4, "Enterprse Lfecycle Investent Manageent, Forth Qarter Reve," brefng, The MITRE Corporaton, Bedford, MA. [3] M. Mollaghase and J. Pet-Edards 997, Makng Mltple-Objectve Decsons, IEEE Copter Socety Press, Los Angeles, CA. [4] R.A. Moynhan and T.N. Sh 4, "Capablty-Based Investent Analyss Spport to Ary G8-FDA Usng the Portfolo AnaLyss MAchne PALMA Tool," MTR 4B88, The MITRE Corporaton, Bedford, MA. [5] J.C. Poerol and S. Barba-Roero, Mltcrteron Decson n Manageent: Prncples and Practce, Kler acadec Pbl., Dordrecht, The Netherlands. [6] H. Raffa 968, Decson Analyss: Introdctory Lectres on Choces nder Uncertanty, Rando Hose, Ne York, NY. [7] B.K. Schdt 7 Eponental Averages, draft MTR, The MITRE Corporaton, Bedford, MA. [8] D. von Wnterfeldt and W. Edards 986, Decson Analyss and Behavoral Research, Cabrdge Unversty Press, Cabrdge, England. 3